De Rham cohomology

In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. It is in different, definite senses dual both to singular homology, and to Alexander-Spanier cohomology.

The differential k-forms on any smooth manifold M form an abelian group (in fact a real vector space) called

Ω^k(M)

under addition. The exterior derivative d gives mappings

d:Ω^k(M) → Ω^k+1(M).

There is a fundamental relationship

d ² = 0;

this follows essentially from symmetry of second derivatives. Therefore vector spaces of k-forms along with the exterior derivative are a cochain complex, the de Rham complex:

In differential geometry terminology, forms which are exterior derivatives are called exact and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms); the relationship d ² = 0 then says that

exact forms are closed.

The cohomology groups of the de Rham complex, which are the vector spaces of closed forms modulo exact forms, are called the de Rham cohomology groups

H^k_DR(M).

The wedge product endows the direct sum of these groups with a ring structure.

De Rham's theorem, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold M, these groups are isomorphic as real vector spaces with the singular cohomology groups

H^p(M;R).

Further, the two cohomology rings are isomorphic (as graded rings).

The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.

Harmonic forms

For a differential manifold M, we can equip it with some auxiliary Riemannian metric. Then the Laplacian Δ, defined by

*d*d + d*d*

using the exterior derivative and Hodge dual defines a homogeneous (in grading) linear differential operator acting upon the exterior algebra of differential forms: we can look at its action on each component of degree p separately.

If M is compact and oriented, the dimension of its kernel acting upon the space of p-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree p: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms, in particular the space of all harmonic p-forms on M is isomorphic to H^p(M;R).